Apparatus and methods for deriving in essentially real time continuous electrical representations of the fourier and inverse fourier transform

ABSTRACT

Apparatus and methods for deriving in essentially real time unweighted and weighted continuous electrical representations of the Fourier transform and/or the inverse Fourier transform of a complex waveform. In performing the Fourier transform, the input waveform is sampled at the Nyquist sampling rate and the samples stored in respective sample-and-hold circuits. These samples are applied to signal generating circuitry for deriving harmonically related time-varying cosine and sine signals having peak values corresponding to weighted or unweighted values of respective ones of the sample-and-hold circuit outputs, and having a fundamental frequency which may be chosen independently of the frequency content of the input waveform. These cosine and sine signals are then respectively summed for producing resultant summed sine and cosine signals which respectively correspond to weighted or unweighted representations of the real and imaginary components of the Fourier transform of the input waveform with the frequency variable being simulated by time. In one embodiment, these summed sine and cosine signals are applied to a function generator for generating signals representative of the weighted or unweighted amplitude spectrum and/or phase spectrum of the input waveform for further application to appropriately calibrated and adjusted oscilloscopes for producing visual displays thereof. In another embodiment, these resultant summed sine and cosine signals are in turn sampled at the Nyquist sampling rate to provide samples which may conveniently be modified in accordance with desired criteria. The modified samples are then recombined using the inverse Fourier transform technique of the invention which employs circuitry basically similar to that used for the Fourier transform to produce an output signal representative of the original input signal and containing the modifications produced in accordance with the desired criteria.

Ol BO-YB (JR 307149566 United States Patent H 1 111 3,714,566

iiiLJ 1.1. .7?

[541 APPARATUS AND METHODS FOR [57] ABSTRACT DERWING IN ESSENTIAL REALApparatussameness is;dent/ing"assess-fewest" TIME CONTINUOUS ELECTRICALtime unweighted and weighted continuous electrical REPRESENTATIONS OFTHE FOURIER representations of the Fourier transform and/or the in- ANDINVERSE FOURIER TRANSFORM verse Fourier transform of a complex waveform.In performin the Fourier transform, the in ut waveform [75] [nvemor'George s"xm,lg"sllver i is sampled at the Nyquist sampling rate and thesam- [73] Assigneel The Bun e C p rat pies stored in respectivesample-and-hold circuits.

Oak Brook,lll. These samples are applied to signal generating cir- [22]Filed: SePL21970 cuit ry for deriving harmonically related time-varyingI cosine and sine signals having peak values correspond- [21] Appl. No.:68,861 ing to weighted or unweighted values of respective ones of thesample-and-hold circuit outputs, and hav- Remed Applicam Dam ing afundamental frequency which may be chosen in- [63] Continuation-in-partof Ser. No. 799,067, Feb. 13, dependently of the frequency content ofthe input 1969. waveform. These cosine and sine signals are thenrespectively summed for producing resultant summed 52 s 324 77 235 15235 131 sine and cosine signals which respectively correspond 324/77 Gto weighted or unweighted representations of the real [5 1} Int. Cl...G01r 23/16 and imaginary Components of the Fourier transform of 581Field of Search ..324/77-, 235/156, 181 the pu Waveform with thefrequency variable being simulated by time. in one embodiment, thesesummed [56] References Cited sine and cosine signals are applied to afunction generator for generating signals representative of the UNITEDSTATES PATENTS weighted or unweighted amplitude spectrum and/or phase sectrum of the input waveform for further ap- 3,2o9,250 9/1965 Burnset al"3 24/77 6 l l g to gppmpiatgly qalibmedl th adjugteld oscr oscopes orpro ucmg vrsua lSp ays ereo. n FOREIGN PATENTS OR APPLICATIONS anotherembodiment, these resultant summed sine and 1,452,084 9/1966 France..235/l8l cosine signals are in turn sampled at the Nyquist samplingrate to provide samples which may conveniently OTHER PUBLICATIONSCochran et al. What is the Fast Fourier Transform in IEEE Transactionson Audio and Electroacoustics Vol. AUl5, No. 2 June 1967 pp. 45-55Primary Examiner-Stanley T. Krawczewicz Att0rneyFrederick M. Arbuckle bemodified in accordance with desired criteria. The

.modified samples are then recombined using the ining the modificationsproduced in accordance with the desired criteria.

47 Claims, 7 Drawing Figures em l 5 8 AMP I ewes, e iuo g 5w EH efinlw egft fl N E I ciRcuiT Z Z c.

9\ i2 5 a 2 a SAMPLE FitTER A. HOLD sw e sinw t Z l (W1) 90 was i 92 n 5a z e 5AM? FlLTER e suuu t' H aQ fi- W) '0 90 Lg H APPARATUS AND METHODSFOR DERIVING IN ESSENTIALLY REAL TIME CONTINUOUS ELECTRICALREPRESENTATIONS OF THE FOURIER AND INVERSE FOURIER TRANSFORMCROSS-REFERENCES TO RELATED PATENT APPLICATIONS s This application is acontinuation-in-part of the commonly assigned copending patentapplication Ser. No. @9967, filed Feb. 13,1969.

This application also contains subject matter generally related to thatcontained in the commonly assigned copending patent application Ser. No.41,363, filed May 28,1970, now U.S. Pat. No. 3,614,673.

BACKGROUND OF THE INVENTION This invention relates to apparatus andmethods for electronically performing the Fourier and inverse Fouriertransforms, and in particular to improved means and methods forderiving, displaying and/or modifying the amplitude and/or phase spectraof an electrical waveform.

As is well known, it is of very considerable value for many types ofapplications to be able to conveniently derive and display the Fourieror inverse Fourier transform of an electrical signal as well as desiredweighted or modified versions thereof. Such capabilities are important,for example, in the study, analysis, equalization and/or utilization ofwaveforms associated with communication and detection systems. The inputelectrical waveform may, for example, be a relatively short pulse, suchas the pulse received by a radar system, or may be a relatively long orcontinuous signal, such as provided by speech, heartbeat, or seismicsignals. Presently known apparatus and methods for deriving, displayingand/or modifying such waveforms have the disadvantage of being undulycomplex and/or requiring an undesirably long time.

BRIEF DESCRIPTION OF THE INVENTION The present invention residesprimarily in the employment of a highly efficient and advantageous novelcombination of digital and analog signal techniques which makes possiblethe derivation, display, and/or modification in essentially real time ofweighted and/or unweighted continuous electrical representations of theFourier and inverse Fourier transfonns of a complex input waveform. Inperforming the Fourier transform in a typical embodiment, the inputwaveform is sampled at the Nyquist sampling rate and the samples storedin respective sample-and-hold circuits. These samples are applied tosignal generating circuitry for deriving harmonically relatedtime-varying cosine and sine signals having peak values corresponding toweighted or unweighted values of respective ones of the sampIe-and-holdcircuit outputs, and having a fundamental frequency which may be chosenindependently of the frequency content of the input waveform. Thesecosine and sine signals are then respectively summed for producingresultant summed sine and cosine signals which respectively correspondto weighted or unweighted representations of the real and imaginarycomponents of the Fourier transform with the frequency variable beingsimulated by time. These resultant summed sine and cosine signals maythen be applied to a function generator for generating continuoussignals in essentially real time which are representative of theweighted or unweighted amplitude spectrum and/or phase spectrum of theinput waveform. Such spectrum signals may advantageously andconveniently be displayed by appropriately calibrated and adjustedoscilloscopes, the period of each displayed spectrum being equal to theperiod of the fundamental frequency used for the derived harmonicallyrelated cosine and sine signals.

A basically similar approach to that described above for performing theFourier transform is also advantageously employed for performing theinverse Fourier transform, except that, in performing the inverseFourier transform the input waveform is provided in the form of twotime-simulated signals corresponding to its real and imaginarycomponents, which signals are separately sampled at the Nyquist rate andthe samples stored in respective sets of sample-and-hold circuits. Thesamples of the real component of the input waveform are used forproviding appropriate peak values for the summed harmonically relatedcosine signals which are generated to form the real component of theinverse Fourier transform,and the samples of the imaginary component ofthe input waveform are used for providing appropriate peak values forthe summed harmonically related sine signals which are generated to formthe imaginary component of the inverse Fourier transform. These real andimaginary inverse Fourier transform components may then be combined toprovide a single continuous signal in essentially real time which isrepresentative of the inverse Fourier transform of the input waveform.

The specific nature of the invention, as well as other features,objects, advantages, and uses thereof, will become apparent from thefollowing description of typical exemplary embodiments taken inconjunction with the accompanying drawings, in which:

FIG. 1 is an electrical block diagram of an exemplary embodiment of theinvention for deriving the Fourier transform of an input waveform foruse in displaying the amplitude and phase spectra thereof;

FIG. 2 is a series of graphs illustrating the outputs of the fixed rateclock 6 and pulse train generator 7 in FIG. 1;

FIG. 3 is an electrical block diagram illustrating details of the add-onunit 55 in FIG. 1;

FIG. 4 is an electrical block diagram illustrating details of thefunction generator 17 in FIG. 1;

FIG. 5 is an electrical block diagram illustrating how a weightingcapability may be provided for the apparatus of FIG. 1;

FIG. 6 is an electrical block diagram of apparatus for deriving aweighted or unweighted electrical representation of the inverse Fouriertransform; and

FIG. 7 is an electrical block diagram illustrating how the apparatus ofFIGS. 1 and 6 may be utilized in a larger system.

Referring to FIG. 1, illustrated therein is an exemplary embodiment ofapparatus for deriving the Fourier transform of an input waveform e(t)for use in displaying the amplitude and phase spectra thereof. The inputwaveform or signal e(t) is applied to an input terminal 1, and, by wayof example, will be assumed to be a received radar pulse. The presenceof this input pulse is sensed, for example, by a conventional form ofthreshold circuit 2 which causes activation of a sampling commutator 3when the pulse is present.

The sampling commutator 3 may typically comprise a ring counter whichoperates during the period of the input pulse to sequentially distributepulses from a variable frequency clock 4 to aplurality of sample-andholdcircuits 5 so as to provide sequential sampling of the input pulse atintervals in accordance with the well known Nyquist sampling theorem. Asis well known, the Nyquist theorem provides that a signal may becompletely specified by samples taken at intervals no greater than l/2W,where W is the highest frequency content of the signal.

Each clock pulse from the variable rate clock 4 acts via the samplingcommutator 3 to momentarily turn on a respective sample-and-hold circuit5 to thereby sample the amplitude of a discrete portion of the inputpulse. Thus, at the end of the input pulse, the outputs of thesample-and-hold circuits, denoted by e, to e,,, will be the respectivesampled amplitude values of the input pulse. It will be understood thatthe total number of sample-and-hold circuits 5 required is N T/t,, whereT is the duration of the input pulse and t, is the sampling interval. Ift is chosen equal to the Nyquist interval l/2W, as will be assumed forthe exemplary embodiment of FIG. 1, then N 2WT.

The sample-and-hold circuits 5 in FIG. 1 may be of conventional form andshould be chosen to have a speed of operation compatible with the inputpulse and the Nyquist sampling interval. It is to be noted that theexemplary embodiment of FIG. 1 permits the number of samples taken onthe input pulse to be readily expanded. The dashed block 50 representsthe basic unit providing for n samples, and the dashed block 55illustrates an add-on unit which may be used to provide for anadditional n sample. Further add-on units may also be provided. Forpresent purposes, it will be assumed that only the basic unit 50 isrequired to meet the Nyquist criterion for the input pulse beinganalyzed, and that n N T/t,. The details of a typical add-on unit willbe considered later on herein in connection with FIG. 3.

From the description so far, it will be understood that the applicationof the input pulse causes samples of the amplitude thereof to be set upin respective ones of the sample-and-hold circuits 5. Since thesesamples are obtained in conformance with the Nyquist sampling thereon,the outputs of the sample-and-hold circuits 5 are sufficient tocompletely specify the input waveform and to thereby permit derivationof the Fourier transform and thus the frequency spectrum therefrom.

The well known mathematical expression for the Fourier transform of aninput signal e(t) is as follows:

The above Fourier transform expression may be written in terms of thesampled values as follows (assuming t,= l/2 W):

where E(w) is the Fourier transform and thus the frequency spectrum ofthe input signal e(t), w is frequency in radians/second; j g V l, e isthe kth sampled value of the input pulse, N is the number of samples,and t, is as previously defined. While there are various possible waysof implementing Equation (1) above, the approach of the presentinvention employs a highly advantageous and remarkably simpleimplementation, as exemplified in FIG. 1, and as will now be described.

For the purposes of the present invention, Equation (1) is expanded andrearranged into its real and imaginary component as follows:

N i N E(w) =t, 6;, cos (lct w) jz k sin 0 5 Y ik=1 k=1 From Equation (2)the amplitude A and phase (1) of the frequency spectrum e(w) areexpressable as:

A V E +E, 3 and where N E 0 cos (M w) and N E 0 sin (kt w With the aboveequations in view, attention is again directed to the exemplaryembodiment of FIG. 1, where it will now be understood that the outputs eto e, of the sample-and-hold circuits 5 respectively correspond to thee,, terms in the above equations. The next step in accordance with theinvention involves the use of these sample-and-hold circuit outputs toderive signals appropriate for use as the terms E and E, of Equations(5) and (6) so as to thereby permit obtaining the desired amplitude andphase spectra using Equations (3) and (4). The manner in which theexemplary embodiment of FIG. 1 provides for the derivation ofappropriate signals for this purpose will next be described.

When the input pulse terminates, samples thereof satisfying the Nyquistcriterion will have been set up on the sample-and-hold circuits 5 inresponse to pulses from the sampling commutator 3. The samplingcommutator 3 next activates a pulse train generator 7 which operates inresponse to pulses received from a fixed rate clock 6 for generating nnumber of harmonically related pulse trains P, to P, on like designatedoutput lines, as illustrated in the graphs of FIG. 2. The pulse traingenerator 7 may be of conventional form and the clock frequency w of theclock 6 used therewith should be much greater than the highest frequencypulse train P,,.

The outputs Pto P, of the pulse train generator 7 are applied torespective switches 8 along with respective outputs e, to e, of thesample-and-hold circuits 5 for v to e,,sinw,,t are obtained by passingthe outputs of the filters 9to 9, through respective 90 phase shifters10. It is to be noted that the above-described means for deriving theterms e,cosw,t to e cosw t and e,sinw,t to e sinw t are relativelysimple and do not require the use of complex multipliers as typicallyfound in conventional spectrum analyzers.

Continuing with the description of FIG. 1, the above derived cosine andsine terms e,cosw,t to e ,,c o s w,,t and e sinw t to e sinw t areindividually summed by application thereof to'respective summingamplifiers 12 for providing the sums it u E :0 cos w t and E e sin w t.k-J k=I As illustrated in FIG. 1, additional summing amplifiers 13 mayalso be provided in the event that more add-on units (such asillustrated by the single add-on unit 55) are required to providesufficient sampling points to satisfy the Nyquist criterion of ZWTsamples. In such a case, the additional summing amplifiers 13 serve torespectively sum the summed cosine and sine terms provided by thesumming amplifiers 12 of the basic unit 50 with the summed cosine andsine terms of other addon units. For example, the summed cosine and sineterms of add-on unit 55 may be expressed as:

I] E Mm. 008 Hu l and n 2 k+u Sin k-Pu k=1 in which case the totalsummed outputs from the basic unit 50 and the add-on unit 55 may beexpressed as follows:

I1 E =2e cos t+a cos w and n E =EG SID wkt+ k=n k+n and the total summedoutputs from the basic unit 50 and all add-on units may be expressed asOf course, if the n samples provided by the basic unit 50 are sufficientto satisfy the Nyquist criterion (i.e., n 2WT), as is being assumed,then the above equations reduce to:

4 t E kzzie cos w (11) and I1 I E,,: c sin w t It is useful to note atthis point in the description that the above derived cosine and sinesummation signals E and E, differ from the terms E and E, of Equations(5) and (6) in that the signals E and E, are time varying rather thanfrequency varying. Nevertheless, in accordance with the invention, thesetime-varying signals E and E, may advantageously be directly used forthe real and imaginary Fourier transform components E and E. inEquations (3) and (4) by using the time variable t to simulate thefrequency variable w; furthermore, the present invention provides thevery considerable advantage that the fundamental frequency used for theharmonically related cosine and sine signals E and E, may be chosenindependently of the frequency content of the input signal e(t). Thechoice of frequencies for E and E, need merely be such that thefrequencies have a harmonic relationship to some desired fundamentalfrequency which may be w, or some lower frequency. In other words, thefrequencies of w, to w, and the fundamental frequency w thereof arechosen to satisfy the following relationships: w, (b l )w,,, w, (b 2)ww, (b n)w,, where b is any desired integer including zero. Since theoperating time required following termination of the input waveform isdependent upon the period of the fundamental frequency used, it will beunderstood that any desired operating time may be provided by properchoice of the fundamental frequency. The reason why such signals as Eand E, can be used for the terms E and E, so as to permit the aboveadvantages to be realized will become evident as the descriptionprogresses.

Since E and E, can be used in place of E and E as pointed out in theprevious paragraph, E and E, can be used in the operations required byEquations (3) and (4). The resulting equations for the amplitude andphase outputs A and 41' are thus expressable as:

and

'=tanE',/E' (bz,)t (14) The additional term (bt,)t in the equation forqb is required in order to take care of the situation when b a 0.

The above operations of equations (13) and (14) are provided in theembodiment of FIG. 1 by feeding the signals E and E, to an appropriatefunction generator 17 along with the output of the fixed rate clock 6.Details of the function generator 17 are shown in FIG. 4 and will beconsidered later on herein. It will be understood that, not only are thetime-varying signals E and E, representing the real and imaginarycomponents of the frequency spectrum much more simply derivable thanwould be signals conforming to the frequency-varying terms E and E,, butalso, the opera tions required by Equations (13) and (I4) using thesignals E and E, can be performed by a much simpler function generatoras compared to the complexity which would be involved in performing theoperations required by Equations (3) and (4) using signals conforming tothe frequency-varying terms E and E,.

The signals A and provided by the function generator 17 are used togenerate displays of the desired amplitude spectrum and phase spectrumby application thereof to respective display means, such as illustratedby cathode ray tube oscilloscopes 18 in FIG. 1, the signals A and (1)each being applied to the vertical input of a respective oscilloscope.It will be understood that a single dual trace oscilloscope may be usedinstead of the two oscilloscopes 18. An oscillograph may also be usedwhere compatible with the speed of response required. The resultingdisplays will be continuous with the horizontal direction correspondingto frequency and with the horizontal width being appropriately chosen toaccommodate the spectra on the oscilloscope screens, each spectrumhaving a period equal to the period of fundamental frequency of thefrequencies w, to w, used for the cosine and sine terms of signals E andE,. Since the horizontal traces of the oscilloscopes 18 can readily beadjusted to accommodate the amplitude and phase spectra for anyfundamental frequency which may be used (within the adjustment range ofthe oscilloscopes), it will be understood that a wide choice isavailable for the fundamental frequency, as pointed out previouslyherein. The frequency calibration in the horizontal direction is suchthat the start of the horizontal trace corresponds to zero frequency andextends in the horizontal direction to a maximum frequency of W at theend of the half period (1r radians) of the fundamental frequency, Wbeing the highest frequency content of the input waveform, as definedpreviously.

It will be understood from the description so far that, if asteady-state display of the amplitude and phase spectra of a singleinput pulse is desired, such a display may be obtained by causing thepulse train generator 7 in FIG. 1 to run continuously after activationby the sampling commutator 3, and by triggering the oscilloscopes 18 atthe fundamental frequency of the cosine and sine terms of signals E andE, at times corresponding to t thereof. Such a trigger signal may bederived, for example, from the pulse train generator 7 in response tothe leading edge of any of the first pulses produced thereby.

If a real-time spectrum analysis is desired on each of a plurality ofpulses provided by the input signal e(t), then the pulse train generator7 is caused to generate pulses for only a single period of thefundamental frequency in response to each cycling by the samplingcommutator 3, which in turn cycles the sample-andhold circuits 5 oncefor each pulse of the input signal e(t). The trigger signal for theoscilloscopes may again be derived from the leading edge of any of thefirst pulses produced by the pulse train generator 7 in FIG. 1. Also,the fundamental frequency of w, to w,, is chosen to provide a shortenough analysis time so that the transient displays of each pulse arecompleted before the next occurring pulse of the input signal. Theresulting transient amplitude and phase spectra appearing on theoscilloscopes 18 may be permanently recorded, for example, using a highspeed camera. Other means for recording may, of course, also beemployed, such as generally illustrated in FIG. 1 by a recorder 19.

Reference is now directed to FIG. 3 for a description of typicalcircuitry which may be employed for the addon unit 55 generallyillustrated in FIG. 1. It will be seen from FIG. 3 that the add-on unit55, similar to the basic unit 50 in FIG. 1, includes n number ofsample-andhold circuits 5 for providing the samples e,, to 2 along withrespective switches 8, narrowband filters 9 to 9,,, phase shifters 10,and two summing amplifiers 12 for deriving the cosine and sine sums l4and two bandpass filters 15, which may be of conventional form, and towhich the outputs of the summing amplifiers 12 are respectively appliedfor performing a frequency up-conversion so as to obtain summed cosineand sine signals 2 m cos wn+t and e sin w t 1.21 is having the properfrequencies for summation in the summing amplifiers 13 in FIG. 1 alongwith the cosine and sine summations from the basic unit 50.

The mixers 14 in FIG. 3 each receive a signal cosw,,t

- from a filter 16 provided in the embodiment of FIG. 1

for mixing with the outputs from the add-on unit summing amplifiers 12to provide mixer output signals M and M, which may be expressed as:

ll F162 .11. ntt 0 wn-k and I1 M =V E e (cos w t+ cos w c t) Thebandpass filters are designed to pass only frequencies within the band Wand to provide a gain of two to compensate for a loss of amplitude inthe mixing operations, as indicated by the l/2"factor in the mixerEquations (19) and Accordingly, the outputs from the filters 15 of theadd-on unit of FIG. 3 will be the desired cosine and sine summationsignals of Equations (17) and (18). It will be understood that otheradd-on units may be constructed in an analogous manner to that shown inFIG. 3, with suitable mixing signals being provided, such as illustratedin FIG. 1 by the provision of filter 16 for use with the add-on unit 55.

Attention is next directed to FIG. 4, which illustrates details of anexemplary function generator which may be employed for the spectrumfunction generator 17 generally illustrated in FIG. 1. As illustrated inFIG. 4, the pulse output at frequency w from the fixed rate clock 6 inFIG. 1 is applied to a filter 20 for producing the sinusoidal signalcosw t which, in turn, is applied to a 90 phase shifter 22 for producingthe quadrature sinusoidal signal sinw t. These signals cosw t and sinw tare applied to respective multipliers 20 along with respective ones ofthe signals E and E, from respective summing amplifiers 13 in FIG. 1 toprovide the multiplier outputs:

E 'cosw' t and E 'sinw t The outputs of the multipliers 20 are summed bya summing amplifier 25 to provide a resultant signal E; expressable as:

and which may be rewritten as:

E,.=. 125 +Eg rcosw zHan- 12,755

It will thus be understood that the envelope of the signal E; at theoutput of the summing amplifier 25 in FIG. 4 is the quantity \IE' E'f,and that the phase difference between the signal E, and the signal coswt from the filter 20 is tan"E,'/E

Accordingly, as illustrated in FIG. 4, by feeding the output signal E,from the summing amplifier 25 to an envelope detector 26 andito a phasedetector 28 (both of which may be of conventional form) along with thesignal cosw t from the filter 20, the signals A and are obtained. For achoice of the fundamental frequency such that w, w,, so that b 0, then(1), in which case the output of the phase detector 28 can be directlyused for 4). However, if a fundamental frequency is chosen so that b isnot zero, the term (bt,,)t must be subtracted from (1V This maytypically be accomplished, as illustrated in FIG. 4, by feeding theoutput signal from the phase detector 28 to a subtrac tor 29 for thepurpose of having the term (bt,)t provided by a sawtooth generator 30subtracted therefrom to produce the resulting desired output signal 4).The subtractor 29 and the sawtooth generator 30 may be of conventionalform, the latter being adjustable in accordance with the chosen valuesfor b and t,.

Referring now to FIG. 5, illustrated therein is a typical manner inwhich the Fourier transform circuitry of FIG. 1 may readily be modifiedto permit a windowweighing capability to be provided. As will beapparent from FIG. 5, the modification merely involves the addition of amultiplier 5a for each frequency channel responsive to a respectivemultiplier factor M, M, and through which the output of the respectivesampleand-hold circuit is fed for introducing a desired weighting withrespect thereto. The outputs E and E, of the summing amplifiers 12 inFIG. 1, which are electrical representations of the real and imaginarycom ponents of the Fourier transform, may then be represented as and n E2 M e sin w t Having described how a weighted or unweighted continuouselectrical representation of the Fourier transform of an input signalmay typically be derived and displayed in essentially real time inaccordance with the invention, next to be considered is the manner inwhich the weighted or unweighted inverse Fourier transform may beperformed in accordance with the invention using a basically similarapproach, as typically illustrated in FIG. 6. The reasons for thissimilarity will become evident as the description progresses. For easeof comparison between FIG. 6 and FIGS. 1 and 5, those elements in FIG. 6which may have similar structures and functions as those in FIGS. 1 and5 are designed by primed numerical designations having the same valuesas their corresponding elements in FIGS. 1 and 5.

It will be seen from FIG. 7 that the input signal E(w) on which theinverse Fourier transform is to be performed is first applied to afunction generator 50, which may be of conventional form, for generatingtime-simulated output signals E,(t) and E,(t) respectively correspondingto the real and imaginery components of the input signal E(w) with thefrequency variable w being simulated by the time variable 2. Often, suchsignals may have already been made available for other purposes in asignal analysis system, so as to thereby obviate having to speciallygenerate them for the inverse Fourier transform approach beingconsidered here. For example, such signals may be available because theanalysis system provides for spectrum analysis using the Fouriertransform derivation approach of the invention described previously.

As will be seen from FIG. 6 these time-simulated real and imaginarysignal components E (t) and E,(t) of the input signal E(w) are eachseparately sampled at a Nyquist rate using a sampling commutator 3,which may be of the same general form as the sampling commutator 3 inFIG. 1, except that it provides for the separate sampling of each of theinput signal real and imaginary components E,(l) and E (t) and thestoring of the resulting samples in appropriate ones of the respectivesets of sample-and-hold circuits provided therefor. The operation of thesampling commutator 3' may, for example, be initiated by a signal 1,,provided by the function generator 50, or else the signal t may be 1derived from some other source, such as a threshold circuit similar tothe threshold circuit 2 in FIG. 1.

At this point in the description of the inverse Fourier transformapproach of the invention, it will be useful to temporarily depart fromthe description of FIG. 6 in order to present the mathematicalrelationships which justify using the values stored in thesample-and-hold circuits 5' for generating a continuous electricalrepresentation of the inverse Fourier transform in essentially real timein a basically similar manner to that used for the Fourier transform.

The well known mathematical expression for the inverse Fourier transforme(t)' of an input signal E(w) is as follows:

By expending EXP[jwt] in the above expression into its cosine and sinefunctions, and by also expanding E(w) into its real and imaginarycomponents E,(w) and E,(w) while restricting the integration interval tothat required for the highest frequency W of interest for e(t), theabove inverse Fourier transform expression may then be expressed asfollows:

In accordance with the invention, it has been found that the aboveEquation (21) for the inverse Fourier transform may advantageously beimplemented in a manner which permits the real and imaginary componentsE,-(w) and E,(w) of the input waveform E(w) to be provided using thesamples stored in the sampleand-hold circuits 5 in FIG. 6 which, aspreviously described, are obtained by a Nyquist sampling of thetime-simulated real and imaginary signals E,(t) and E,(t). Accordingly,Equation (21 may be converted to the following form for use inperforming the inverse Fourier transform in accordance with theinvention:

N N u Elf. cos (Mam-2 E sin (lmw) k l k=1 where E is the kth sampledvalue of the time-simulated real component signal E (t), E is the kthsampled value of the time-simulated imaginary component signal e,( t), Nis the number of samples, and t,(= AW/ 1r) is the sampling intervalemployed between samples taken on each of E,(t) and E,(t). The primedsymbol N and t, are used in Equation (22) to distinguish from theunprimed symbols N and t, used in Equation (2) in connection with theFourier transform.

The number of samples N used in implementing the above Equation (22)should, of course, be sufficient to obtain a resulting inverse transformsignal e(t)' which accurately represents the desired inverse Fouriertransform of the input signal E(w). Accordingly, N may be chosensufficient to satisfy the Nyquist criteria based on the frequencycontents of E,(t) and EU). N may also be chosen based on an examinationof e(t) to make sure that N is at least large enough so that theenvelope of the resulting inverse Fourier transform signal e(t)' decaysto essentially zero magnitude between successive periods.

Returning now to the inverse Fourier transform circuitry illustrated inFIG. 6, it will now be evident from Equation (22) and the previousdiscussion that the samples in the respective sample-and-hold circuits5' may advantageously be employed in cooperation with multipliers 5a,pulse train generator 7, switches 8, filters 9 to 9,, and summingamplifiers 12' to synthesize the harmonically related cosine and sinefunctions of appropriate peak amplitudes required by the inverse Fouriertransform in the same basic way as previously described in connectionwith FIGS. 1 and 5 for the Fourier transform. However, since the peakamplitudes of the cosine and sine functions of the inverse Fouriertransform are not the same as they are for the Fourier transform, theinverse Fourier transform circuitry of FIG. 6 differs from FIG. 1 inthat it provides for the separate synthesizing of both the sine andcosine functions, rather than using the FIG. I approach in which thesine functions are obtained by a 90 phase-shifting of the cosinefunctions. Accordingly, the pulse train generator 7 of FIG. 6 willdiffer from the pulse train generator 7 of FIG. 1, since it willadditionally have to provide for the separate synthesizing of the sinefunctions by the provision of the additional signals P, to P,,' having90 phase shifts relative to respective ones of the signals P to P,,.

It will thus be apparent from FIG. 6 that the respective output signalsfrom the summing amplifiers 12 will constitute weighted or unweightedcontinuous electrical representations in essentially real time of thecomponents making up Equation (22), and may accordingly be applied to asubtracting amplifier 52 for providing a resulting electricalrepresentation of the desired inverse Fourier transform e(t)'. As can bedone for the Fourier transform, the fundamental frequency w of thesynthesized harmonically related cosine and sine functions used for theinverse Fourier transform may also be chosen to have any desired value,depending on the use intended for the resulting inverse Fouriertransform signal e(t)'. Also, the signal e(t) may be applied to aconventional frequency translator to provide a like signal e(t) exceptshifted to a desired frequency band.

Having described how weighted and unweighted electrical representationsof the Fourier and inverse Fourier transform may be provided inaccordance with the invention, reference is next directed to FIG. 7which illustrates how both may be utilized in a larger system involvingthe derivation and modification of the frequency spectrum of an inputwaveform e(t) in accordance with desired criteria.

As will be seen from FIG. 7, an input signal e(t) whose frequencyspectrum is to be modified is applied to the Fourier transform circuitryof FIG. 1 for producing output signals E and E, respectivelycorresponding to the real and imaginary components of the Fouriertransform of e(t), which are thus representative of .the frequency andphase spectra of e(t), as will be evident from the previous descriptionin connection with FIG. I. It will, of course, be understood that thesesignals E and E, indicated in FIG. 7 are those appearing at the outputsof the summing amplifiers 12 in FIG. 1. In accordance with theembodiment of the invention illustrated in FIG. 7, these signals E andE, obtained from the Fourier transform circuitry of FIG. I areadvantageously used as the time-simulated real and imaginary componentsE,(t) and E,(t) required for application to the inverse Fouriertransform circuitry illustrated in FIG. 6. As will shortly becomeapparent, the inverse Fourier transform circuitry of FIG. 6 is employedin the embodiment of FIG. 7 for modifying the frequency spectrum (and/orphase spectrum) of e(t) represented by E, and E, in accordance withdesired criteria, and then generating an electrical output signal e(r)'vrepresentative of the modified input waveform For example, theembodiment of FIG. 7 could be used for the purpose of line equalization.In such an application, e(t) moved to an impulse response of a line, ande(t)' would be the impulse response of the equalizer. The equalizationrequirement would then be that the amplitude spectrum of the product ofe(t) and e(t)' be flat within the frequency band of interest, and thatthe phase spectrum of the product of e(t) and e(t)' be a linear functionof frequency. The reference DC values in FlG. 7 would then be sampledvalues of the Fourier spectrum of an ideal line impulse response, andEU) and E,(t) would respectively be the real and imaginary parts of thefrequency response of the actual line, where the frequency variable issimulated by the time variable t. The output -e(t)' would then be animpulse response of the equalizer, which may be sampled to set the tapgains of a transversal filter in order to provide the desiredequalization.

Accordingly, referring to FIG. 6 along with FIG. 7, it will be seen thatthe samplings E,, to E and E to E obtained from the outputs of thesampleand-hold circuits 5' in FIG. 6 are applied to a comparator 85 inFIG. 7 for comparison with respective corresponding reference DC valuesR, to R and R to R which, for example, might represent a referenceFourier spectrum. The comparator 85 may be of conventional form forproviding this comparison in accordance with any desired criteria thatone might wish to provide. The outputs from the comparator 85 areapplied to an arithmetic unit 80 which operates in response thereto toprovide values for the weighting factors M, to M and M to M applied tothe multipliers 5a of the inverse Fourier transform circuitry of FIG. 6,whereby to provide the desired form for the resulting output signal e(t)in accordance with the results of the comparison. For example, if thecriteria provided for comparison by the comparator 85 is equalitybetween each of the respective values of the sample-and-hold circuitoutputs E,, to E and E to E and the reference DC values R to R and R toR then the arithmetic unit 80 will operate to provide weighting factorsM,., to M and M to M, which will cause the circuitry of FIG. 6 toproduce a resulting output signal e(t)' satisfying these equalities. Itwill be appreciated that the capability of being able to perform thedesired comparison using the sample-and-hold circuit outputs and the DCreference values, rather than having to compare complex waveforms, is ofvery considerable advantage with regard to simplicity, economy andefficiency. Of course, if the desired modifications are known inadvance, the comparison operation indicated in FIG. 7 could be omittedand the arithmetic unit pre-set to provide weighting factors which willproduce the desired form for the resultant output signal e(t)'.

As a final point with respect to the system of FIG. 7, it will beappreciated that the particular frequency band of the resulting outputsignal e(t)" will be dependent upon the harmonically related frequenciesw to w,, chosen to be synthesized by the inverse Fourier transformcircuitry of FIG. 6 and the frequency translation provided by thefrequency translator 70. These could, of course, be chosen so that thefrequency band of e(t)" is the same as that of the original input signale(t).

Although the description herein has been primarily concerned withparticular exemplary embodiments of the invention, it is to beunderstood that the invention is not limited thereto, and that a widevariety of modifications and variations in construction, arrangement,method of operation and/or usage may be made without departing from thescope of the invention. For example, it will be apparent to thoseskilled in the art from the description provided herein that desiredweightings of the Fourier and inverse Fourier transforms may be providedat various other points and in various other ways different from thatshown in the described embodiments. The present invention is thusproperly to be considered as including all possible means and methodsfor carrying out the invention coming within the scope of the appendedclaims.

The embodiments of the invention in which an exclusive property orprivilege is claimed are defined as follows:

1. In apparatus for deriving an electrical signal representative of aFourier-type mathematical transformation of an input signal, thecombination of:

sampling means including a plurality of sample-andhold circuits forproviding amplitude sampling of said input signal and for simultaneouslystoring the resulting amplitude samples of said input signal inrespective ones of said sample-and-hold circuits,

generating means responsive to the samples stored in saidsample-and-hold circuits for simultaneously generating harmonicallyrelated time-varying cosine and sine signals having peak valuesproportional to respective ones of said samples, and

means for simultaneously combining respective ones of said cosine andsine signals to produce resultant electrical signals representative ofthe desired Fourier-type mathematical transformation.

2. The invention in accordance with'claim 1,

wherein said desired Fourier-type transform is the Fourier transform ofthe input signal, and

wherein said sampling means operates to sample said input signal atleast at a Nyquist sampling rate to provide amplitude samples thereofwhich are.

stored in respective ones of said sample-and-hold circuits.

3. The invention in accordance with claim 2 wherein said time-varyingharmonically related cosine and sine signals are expressable ase,cosw,t, e cosw t, e,,cosw,,t and e sinw t, e sinw t e,,- cosw t, where2 e 2,, correspond to said samples, n is the number of samples, t is thetime variable, and w is frequency in radians per second, and

wherein the frequencies w to w,, and the fundamental frequency w,thereof satisfy the relationships w (b=l )w w (b+2)w,,, w,,= (b-l-n)w,,,where b is an integer and includes zero.

4. The invention in accordance with claim 3,

wherein said input signal comprises a plurality of pulses, and

wherein the fundamental frequency w is chosen to provide an analysistime less than the time between pulses.

5. The invention in accordance with claim 3 wherein the fundamentalfrequency w of said cosine and sine signals may be chosen independentlyof the frequency content of said input signal.

6. The invention in accordance with claim 1,

wherein said desired Fourier-type transform is the inverse Fouriertransform of the input signal,

wherein said input signal is provided in the form of two time-simulatedsignals corresponding to its real and imaginary components,

wherein a separate set of sample-and-hold circuits is provided for theamplitude samples of each of said time-simulated signals,

wherein said sampling means operates to sample each of saidtime-simulated signals at least at a Nyquist sampling rate to provideamplitude samples thereof which are stored in respective ones of therespective sets of sample-and-hold circuits, and

wherein said generating means responds to said amplitude samples in amanner so that the harmonically related time-varying cosine signals havepeak values proportional to respective ones of the amplitude samplesobtained from the time-simulated signal corresponding to the realcomponent of the input signal and so that the harmonically relatedtime-varying sine signals have peak values proportional to respectiveones of the amplitude samples obtained from the time-simulated signalcorresponding to the imaginary component of the input signal.

7. The invention in accordance with claim 6 wherein said time-varyingharmonically related cosine and sine signals are expressable ase,cosw,t, e,cosw,t, e,,cosw,,t and e,sinw,t, e,sinw,t 2,,- cosw t, wheree e e, correspond to said samples, n is the number of samples, I is thetime variable, and w is frequency in radians per second, and

wherein the frequencies w to w,, and the fundamental frequency w,thereof satisfy the relationships w (b+l )w,,, w: (b+2)w,,, w,(b-l-n)w,,, where b is an integer and includes zero.

8. The invention in accordance with claim 7 wherein the fundamentalfrequency w,, of said cosine and sine signals may be chosenindependently of the frequency content of said input signal.

9. ln apparatus for performing Fourier-type mathematical manipulationsof electrical signals in essentially real-time, the combination of:

means for deriving time-simulated electrical signals representative ofthe real and imaginary components of the Fourier transform of an inputsignal,

means for providing separate Nyquist sampling of each of the real andimaginary component signals and for simultaneously providing fixedamplitude samples thereof,

means for modifying said amplitude samples in accordance with desiredcriteria, and

generating means responsive to the modified samples for performing aninverse Fourier transformation to produce essentially real-timeresultant electrical signals representative of the real and imaginarycomponents of said input signal but containing the modificationsintroduced by said means for modifying.

10. The invention in accordance with claim 9 wherein said generatingmeans includes:

means for simultaneously generating time-varying harmonically relatedcosine and sine signals having peak values proportional to respectiveones of said modified samples, and

means for simultaneously summing respective ones of said cosine and sinesignals to produce said resultant electrical signals.

11. In apparatus for performing Fourier-type mathematical manipulationson electrical signals, the combination of:

first sampling means including a first plurality of sample-and-holdcircuits for sampling an input signal and for storing amplitude samplesrepresentative thereof in respective ones of said first plurality ofsample-and-hold circuits,

first generating means responsive to said sampling means for generatinga first series of harmonically related time-varying cosine and sinesignals having peak values proportional to respective ones of saidsamples,

first combining means for respectively summing and first series ofcosine and sine signals to produce time-simulated electrical signalsrepresentative of the real and imaginary components of the Fouriertransform of said input signal,

second sampling means including second and third pluralities ofsample-and-hold circuits for separately sampling each of the real andimaginary component signals and for storing amplitude samplesrepresentative thereof in respective ones of said second and thirdpluralities of sample-andhold circuits,

means for modifying the samples stored in said second and thirdpluralities of sample-and-hold circuits in accordance with desiredcriteria,

second generating means responsive to the modified samples forgenerating a second series of harmonically related time-varying cosineand sine signals having peak values proportional to respective ones ofthe modified samples derived from said real component signal, and

second combining means for respectively summing the second series ofcosine and sine signals.

12. The invention in accordance with claim 1 l,

where wherein said means for modifying includes means for comparing thesamples stored in said second and third pluralities of sample-and-holdcircuits with predetermined reference values and for modifying sandsamples in response to the results of the comparison.

13. The invention in accordance with claim 11,

wherein said apparatus includes third combining means for combining therespectively summed second series of cosine and sine signals to producean electrical signal representative of said input signal but containingthe modifications introduced by said means for modifying.

14. In apparatus for analyzing a waveform,

means for simultaneously providing amplitude samples of said waveform atintervals of t, l/2W, where z, is the interval between samples and W isthe highest frequency content of said waveform,

means for simultaneously producing time-varying harmonically relatedcosine and sine signals having peak values respectively proportional tosaid samples and having a fundamental frequency which is independent ofthe frequency content of said waveform, and

means for simultaneously combining respective ones of said cosine andsine signals to produce a timevarying spectrum signal having a timevariable simulating the frequency variable of the spectrum of saidwaveform and having a period equal to the fundamental frequency of saidtime-varying harmonically related cosine and sine signals.

15. The invention in accordance with claim 14,

wherein said time-varying harmonically related cosine and sine signalsare expressable as e cosw t, e cosw t, e,,cosw,,t and e sinw t, e sinw te cosw t, where e e e, correspond to said samples, n is the number ofsamples, t is the time variable, and w is frequency in radians persecond, and

wherein the frequencies w, to w, and the fundamental frequency w,thereof satisfy the relationships w (b+l )w w (b+2)w,,, w (b+n)w,,,where b is an integer and includes zero.

16. The invention in accordance with claim 14,

wherein said spectrum signal corresponds to the amplitude spectrum ofsaid waveform and is expressable as and ll E 2 C sin w t 17. Theinvention in accordance with claim 15, wherein said spectrum signalcorresponds to the phase spectrum of said waveform and is expressable asch =tan E,,/E

where 11 1' E :6 cos w i and r miss? and where n E =E e cos w t and nE.=2 e sin w t 19. The invention in accordance with claim 28,

wherein said apparatus additionally includes output means to which saidspectrum signal is applied for providing a manifestation of saidwaveform.

20. The invention in accordance with claim 19 wherein said output meansincludes cathode ray tube means for producing a display of at least oneof the amplitude and phase spectra of said input waveform.

21. The invention in accordance with claim 14 wherein said waveformcomprises a plurality of pulses, and

wherein said fundamental frequency is chosen to provide an analysis timeless than the time between said pulses.

22. The invention in accordanc e with claim 14,

wherein said means for producing time-varying harmonically relatedcosine and sine signals includes means for producing a first group oftime-varying harmonically related cosine and sine signals having peakvalues respectively corresponding to a first plurality of said samples,

means for producing a second group of time-varying harmonically relatedcosine and sine signals of like frequencies of those of said first groupand having amplitudes corresponding to a second plurality of samples,and

means for converting the frequencies of the second group of cosine andsine signals to frequencies which continue the harmonic relation of thefirst group.

23. The invention in accordance with claim 18, wherein said means forcombining includes:

means for producing time-varying cosine and sine signals of likefrequency,

means for multiplying the signal E, by one of the cosine and sinesignals of like frequency and for multiplying the signal E, by theother,

means for summing the thus multiplied signals, and

envelope detector means for producing the amplitude spectrum signal Afrom the summed multiplied signals.

24. The invention in accordance with claim 19,

wherein said means for combining additionally includes:

phase detector means to which the summed multiplied signals are appliedalong with one of said cosine and sine signals of like frequency fordetecting the phase difference 4),, therebetween, and

means for subtracting the term (bt,)t from 45,, for

producing the phase spectrum signal 25. In apparatus for analyzing awaveform,

a basic unit for providing a plurality of samples of said waveform atintervals satisfying the Nyquist sampling theorem and for producingtime-varying harmonically related cosine and sine signals having peakvalues respectively proportional to said sampics,

at least one add-on unit for providing an additional plurality ofsamples of said waveform at Nyquist sampling intervals and for producingtime-varying harmonically related cosine and sine signals having peakvalues respectively proportional to the additional plurality of samplesand with like frequencies as those produced by said basic unit,

said add-on unit also including means for up-converting the frequenciesof the cosine and sine signals produced thereby to frequencies such thatthe cosine and sine signals of the add-on unit will continue theharmonic relation of the cosine and sine signals of the basic unit, and

means for combining all of said cosine and sine signals to produce atime-varying spectrum signal of said waveform with the time variable ofsaid spectrum signal simulating the frequency variable of the spectrumof said waveform and said spectrum signal having a period equal to thefundamental frequency of the time-varying harmonically related sine andcosine signals.

26. The invention in accordance with claim 25,

wherein the cosine and sine signals produced by said basic unit areexpressable as e cosw t, e cosw t e,,cosw,,t and e sinw t, e sinw te,,sinw,,t, where e e e,, are the samples provided by said basic unit, wto w, are the harmonically related frequencies of the cosine and sinesignals, and n is the number of samples provided by the basic unit,

wherein the cosine and sine signals produced by said add-on unit priorto application to said means for up-converting are expressable as e coswt, e t cos t, .e,, ,,,cosw,,t and e,, sinw,t, e sinw ,t e,, ,,,sinw,,t,where e e,, e,, are the samples provided by said add-on units, w to w,are as defined above, and m is the number of samples provided by theadd-on unit,

wherein the cosine and sine signals provided by said add-on unit afterapplication to said means for upconverting are expressable as e,,,cosw,, e cos w t e,, ,,,cosw,, ,,,t, and e,, ,sinw,, ,t, e sinw 1 e,,,,,sinw,, ,,,t, and

wherein the frequencies w to w,, and w to w,,

and the fundamental frequencies w, thereof satisfy the relationships w(b+l )w,,, w, (b+2)w m m m r|+2 2)w w,, (b+n+m)w,,, where b is aninteger including zero.

27. In a method of analyzing the spectrum of a waveform in essentiallyreal-time, the steps of:

amplitude sampling said waveform at intervals oft,

l/2W, where t, is the interval between samples and W is the highestfrequency content of said waveform,

simultaneously generating time-varying harmonically related cosine andsine signals having peak values 5 respectively proportional to saidsamples and expressable as e cosw t, e cosw t e,,cosw,,t and e; sin wt,e Slllwzl e Sll'l w t,where 6 ,82 e

correspond to said samples, n is the number of samples, t is the timevariable and w is frequency in radians per second, and the frequenciesw, to w,,

and the fundamental frequency w thereof satisfy fl w/Tm?? and =tan E /E(bt )t where n 136:2 ek cos w t' k=1 and 11 E.=E e sin w t 29. Theinvention in accordance with claim 27,

wherein the step of generating is performed by steps includinggenerating pulse trains having harmonically related frequencies withamplitudes proportional to said samples, and

filtering the resultant signals over narrow frequency bandscorresponding to the respective frequencies of said pulse trains. 30.The invention in accordance with claim 27, wherein said steps ofsampling and generating are performed by steps including generating afirst group of time-varying harmonically related cosine and sine signalshaving amplitudes proportional to a first plurality of said samples,

generating a second group of time-varying harmonically related cosineand sine signals of like frequencies as those of said first group andhaving amplitudes proportional to a second plurality of said samples,and

converting the frequencies of the second group of cosine and sinesignals to frequencies which continue the harmonic relation of the firstgroup.

31. in a method of deriving an essentially real-time electrical signalrepresentative of a desired Fouriertype mathematical transformation ofan input signal, the steps of:

sampling said input signal to provide fixed value amplitude samplesrepresentative thereof,

simultaneously generating harmonically related time-varying cosine andsine signals having peak values proportional to respective ones of saidsamples, and

simultaneously combining said cosine and sine signals to produceessentially real-time resultant electrical signals representative of thedesired Fourier-type mathematical transformation.

32. The invention in accordance with claim 31,

wherein said desired Fourier-type mathematical transformation is theFourier transform of the input signal,

wherein said sampling is such as to sample said input waveform in amanner so that said samples are amplitude samples of said input signaltaken at intervals satisfying the Nyquist sampling rate, and

wherein said resultant electrical signals comprise two time-simulatedelectrical signals representative of the real and imaginary componentsof the Fourier transform of the input signal.

33. The invention in accordance with claim 31 wherein said desiredFourier-type mathematical transformation is the inverse Fouriertransform of the input signal,

wherein said input signal is provided in the forms of time-simulatedsignals corresponding to its real and imaginary components, and

wherein said sampling is such as to separately sample each of saidtime-simulated signals in a manner so that said samples are amplitudesamples thereof taken at sufficiently closely spaced intervals so thatsaid resultant electrical signals are representative of the inverseFourier transform.

34. The invention in accordance with claim 33,

wherein said generating is such that the harmonically relatedtime-varying cosine signals having peak values proportional torespective ones of the amplitude samples obtained from thetime-simulated signal corresponding to the real component of the inputsignal and the harmonically related timevarying sine signals have peakvalues corresponding to respective ones of the amplitude samplesobtained from the time-simulated signal corresponding to the imaginarycomponent of the input signal.

35. In a method of performing Fouriertype mathematical manipulations onelectrical signals, the steps of:

sampling an input signal at least at a Nyquist sampling rate to providefixed value amplitude samples representative thereof,

generating a first series of harmonically related timevarying cosine andsine signals having peak values proportional to respective ones of saidsamples,

combining said cosine and sine signals to produce time-simulatedelectrical signals representative of the real and imaginary componentsof the Fourier transform of said input signal,

sampling each of the real and imaginary component signals at least at aNyquist sampling rate to provide fixed value amplitude samples thereof,

modifying the fixed value amplitude samples obtained from sampling thereal and imaginary component signals in accordance with desiredcriteria,

generating a second series of harmonically related time-varyingcosineand sine signals having peak values proportional to respectiveones of the modified samples, and

respectively summing the second series of cosine and sine signals.

36. The invention in accordance with claim 35,

wherein said method includes the step of comparing the fixed valueamplitude samples obtained from sampling the real and imaginarycomponent signals with predetermined reference values, and wherein thestep of modifying modifies the amplitude samples obtained from samplingthe real and imaginary component signals in response to' the resultsobtained during the step of comparing. 37. The invention in accordancewith claim 35, wherein said method includes the step of combining therespectively summed second series of cosine and sine signals to producea resultant electrical signal representative of said input signal butcontaining the modifications introduced by the step of modifying. 38. Inapparatus for deriving an electrical signal representative of aFourier-type mathematical transformation of an input signal, thecombination of:

sampling means including a plurality of sample and hold circuits forsampling said input signal and for storing samples representativethereof in respective ones of said sample-and-hold circuits,

generating means responsive to the samples stored in saidsample-and-hold circuits for generating harmonically relatedtime-varying cosine and sine signals having peak values proportional torespective ones of said samples, said generating means including:

means for producing a first group of time-varying harmonically relatedcosine and sine signals having peak values respectively corresponding toa first plurality of said samples,

means for producing a second group of time-varying harmonically relatedcosine and sine signals of like frequencies of those of said first groupand having amplitudes corresponding to a second plurality of samples,

means for converting'the frequencies of the second group of cosine andsine signals to frequencies which continue the harmonic relation of thefirst group, and

means for combining said cosine and sine signals to produce resultantelectrical signals representative of the desired Fourier-typemathematical transformation.

39. In apparatus for deriving an electrical signal representative of aFourier transform of an input signal, the combination of:

sampling means including a plurality of sample-andhold circuits forsampling said input signal at least at a Nyquist rate and for storingamplitude samples representative thereof in respective ones of saidsample-and-hold circuits,

generating means responsive to the samples stored in saidsample-and-hold circuits for generating harmonically relatedtime-varying cosine and sine signals having peak values proportional torespective ones of said samples, said time-varying harmonically relatedcosine and sine signals being expressable as e coswfl, e cosw tf.e,,cosw,,t and e sin 10 1,62 sin 1112i e sin w t, where 6 ,62 ecorrespond to said samples, n is the number of samples, t is the timevariable, and w is frequency in radians per second, and where thefrequencies w to w, and the fundamental frequency w thereof satisfy therelationships w, (b+l )w w (b+2 )w w, (b+n)w wherein b is an integer andincludes zero, and where the fundamental frequency w of said cosine andsine signals may be chosen independently of the frequency content ofsaid input signal, and

means for combining said cosine and sine signals to produce resultantelectrical signals representative of the desired Fourier-typemathematical transformation, said combining means operating torespectively sum said cosine and sine signals so that said resultantsignals comprise two signals corresponding to the real and imaginarycomponents of the Fourier transform of the input signal.

40. The invention in accordance with claim 39,

wherein said generating means includes means for weighting the peakvalues of said cosine and sine signals.

41. The invention in accordance with claim 39,

wherein said combining means includes means for combining said resultantsignals to provide a timevarying spectrum signal representative of theamplitued spectrum of the input signal with the time variable simulatingthe frequency variable and having a period equal to the fundamentalfrequency w, of said cosine and sine signals.

42. The invention in accordance with claim 41,

wherein said combining means additionally includes means for combiningsaid resultant signals to provide a continuous time-varying spectrumsignal representative of the phase spectrum of the input signal with thetime variable simulating the frequency variable of the phase spectrumand having a period equal to the fundamental frequency w,, of saidcosine and sine signals.

43. The invention in accordance with claim 42,

wherein said apparatus includes means for displaying at least one of theamplitude and phase spectra signals.

44. In apparatus for deriving an electrical signal representative of aninverse Fourier transform of an input signal provided in the form of twotime-simulated signals corresponding to its real and imaginarycomponents, the combination of:

hold circuits for sampling said time-simulatedsignals at least at aNyquist rate and for storing amplitude samples representative thereof inrespective ones of said sample-and-hold circuits, a separate set ofsample-and-hold circuits being provided for the amplitude samples ofeach of said time-simulated signals,

generating means responsive to the samples stored in saidsample-and-hold circuits for generating harmonically relatedtime-varying cosine and sine signals having peak values proportional torespective ones of said samples, said time-varying harmonically relatedcosine and sine signals being expressa ble as e cosw t, e cosw t, e coswt and rSlIlW1T,zSll'lW2I. encosw t. where e1, e2 correspond to saidsamples, n is the number of samples, I is the time variable, and w isfrequency In radians per second, and where the frequencies w to w, andthe fundamental frequency w thereof satisfy the relationships w =(b+l)w,,, w (b+2 )w w (b+n)w where b is an integer and includes zero, andwhere the fundamental frequency w of said cosine and sine signals may bechosen independently of the frequency content of said input signal, and

means for combining said cosine and sine signals to produce resultantelectrical signals representative of the desired Fourier-typemathematical transformation, said combining means operating torespectively sum said cosine and sine signals so that said resultantsignals comprise two signals corresponding to the real and imaginarycomponents of the inverse Fourier transform of the input signal.

45. The invention in accordance with claim 44,

wherein said generating means includes means for weighting the peakvalues of said cosine and sine signals.

46. The invention in accordance with claim 45,

wherein said combining means includes means for combining said resultantsignals to provide a single continuous time-varying signalrepresentative of the inverse Fourier transform of the input signal.

47. The invention in accordance with claim 46,

wherein said apparatus includes means for providing a frequencytramslation of said single time-varying signal.

1. In apparatus for deriving an electrical signal representative of aFourier-type mathematical transformation of an input signal, thecombination of: sampling means including a plurality of sample-and-holdcircuits for providing amplitude sampling of said input signal and forsimultaneously storing the resulting amplitude samples of said inputsignal in respective ones of said sample-and-hold circuits, generatingmeans responsive to the samples stored in said sample-and-hold circuitsfor simultaneously generating harmonically related time-varying cosineand sine signals having peak values proportional to respective ones ofsaid samples, and means for simultaneously combining respective ones ofsaid cosine and sine signals to produce resultant electrical signalsrepresentative of the desired Fourier-type mathematicaltransformation.
 1. In apparatus for deriving an electrical signalrepresentative of a Fourier-type mathematical transformation of an inputsignal, the combination of: sampling means including a plurality ofsample-and-hold circuits for providing amplitude sampling of said inputsignal and for simultaneously storing the resulting amplitude samples ofsaid input signal in respective ones of said sample-and-hold circuits,generating means responsive to the samples stored in saidsample-and-hold circuits for simultaneously generating harmonicallyrelated time-varying cosine and sine signals having peak valuesproportional to respective ones of said samples, and means forsimultaneously combining respective ones of said cosine and sine signalsto produce resultant electrical signals representative of the desiredFourier-type mathematical transformation.
 2. The invention in accordancewith claim 1, wherein said desired Fourier-type transform is the Fouriertransform of the input signal, and wherein said sampling means operatesto sample said input signal at least at a Nyquist sampling rate toprovide amplitude samples thereof which are stored in respective ones ofsaid sample-and-hold circuits.
 3. The invention in accordance with claim2 wherein said time-varying harmonically related cosine and sine signalsare expressable as e1cosw1t, e2cosw2t, ... encoswnt and e1sinw1t,e2sinw2t . . . encoswnt, where e1, e2, . . . en correspond to saidsamples, n is the number of samples, t is the time variable, and w isfrequency in radians per second, and wherein the frequencies w1 to wnand the fundamental frequency wo thereof satisfy the relationships w1 (b1)wo, w2 (b+2)wo, . . . wn (b+n)wo, where b is an integer and includeszero.
 4. The invention in accordance with claim 3, wherein said inputsignal comprises a plurality of pulses, and wherein the fundamentalfrequency wo is chosen to provide an analysis time less than the timebetween pulses.
 5. The invention in accordance with claim 3 wherein thefundamental frequency wo of said cosine and sine signals may be chosenindependently of the frequency content of said input signal.
 6. Theinvention in accordance with claim 1, wherein said desired Fourier-typetransform is the inverse Fourier transform of the input signal, whereinsaid input signal is provided in the form of two time-simulated signalscorresponding to its real and imaginary components, wherein a separateset of sample-and-hold circuits is provided for the amplitude samples ofeach of said time-simulated signals, wherein said sampling meansoperates to sample each of said time-simulated signals at least at aNyquist sampling rate to provide amplitude samples thereof which arestored in respective ones of the respective sets of sample-and-holdcircuits, and wherein said generating means responds to said amplitudesamples in a manner so that the harmonically related time-varying cosinesignals have peak values proportional to respective ones of theamplitude samples obtained from the time-simulated signal correspondingto the real component of the input signal and so that the harmonicallyrelated time-varying sine signals have peak values proportional torespective ones of the amplitude samples obtained from thetime-simulated signal corresponding to the imaginary component of theinput signal.
 7. The invention in accordance with claim 6 wherein saidtime-varying harmonically related cosine and sine signals areexpressable as e1cosw1t, e2cosw2t, . . . encoswnt and e1sinw1t, e2sinw2t. . . encoswnt, where e1, e2, . . . en correspond to said samples, n isthe number of samples, t is the time variable, and w is frequency inradians per second, and wherein the frequencies w1 to wn and thefundamental frequency wo thereof satisfy the relationships w1 (b+1)wo,w2 (b+2)wo, . . . wn (b+n)wo, where b is an integer and includes zero.8. The invention in accordance with claim 7 wherein the fundamentalfrequency wo of said cosine and sine signals may be chosen independentlyof the frequency content of said input signal.
 9. In apparatus forperforming Fourier-type mathematical manipulations of electrical signalsin essentially real-time, the combination of: means for derivingtime-simulated electrical signals representative of the real andimaginary components of the Fourier transform of an input signal, meansfor providing separate Nyquist sampling of each of the real andimaginary component signals and for simultaneously providing fixedamplitude samples thereof, means for modifying said amplitude samples inaccordance with desired criteria, and generating means responsive to themodified samples for performing an inverse Fourier transformation toproduce essentially real-time resultant electrical signalsrepresentative of the real and imaginary components of said input signalbut containing the modifications introduced by said means for modifying.10. The invention in accordance with claim 9 wherein said generatingmeans includes: means for simultaneously generating time-varyingharmonically related cosine and sine signals having peak valuesproportional to respective ones of said modified samples, and means forsimultaneously summing respective ones of said cosine and sine signalsto produce said resultant electrical signals.
 11. In apparatus forperforming Fourier-type mathematical manipulations on electricalsignals, the combination of: first sampling means including a firstplurality of sample-and-hold circuits for sampling an input signal andfor storing amplitude samples representative thereof in respective onesof said first plurality of sample-and-hold circuits, first generatingmeans responsive to said sampling means for generating a first series ofharmonically related time-varying cosine and sine signals having peakvalues proportional to respective ones of said samples, first combiningmeans for respectively summing and first series of cosine and sinesignals to produce time-simulated electrical signals representative ofthe real and imaginary components of the Fourier transform of said inputsignal, second sampling means including second and third pluralities ofsample-and-hold circuits for separately sampling each of the real andimaginary component signals and for storing amplitude samplesrepresentative thereof in respective ones of said second and thirdpluralities of sample-and-hold circuits, means for modifying the samplesstored in said second and third pluralities of sample-and-hold circuitsin accordance with desired criteria, second generating means responsiveto the modified samples for generating a second series of harmonicallyrelated time-varying cosine and sine signals having peak valuesproportional to respective ones of the modified samples derived fromsaid real component signal, and second combining means for respectivelysumming the second series of cosine and sine signals.
 12. The inventionin accordance with claim 11, wherein said means for modifying includesmeans for comparing the samples stored in said second and thirdpluralities of sample-and-hold circuits with predetermined referencevalues and for modifying sand samples in response to the results of thecomparison.
 13. The invention in accordance with claim 11, wherein saidapparatus includes third combining means for combining the respectivelysummed second series of cosine and sine signals to produce an electricalsignal representative of said input signal but containing themodifications introduced by said means for modifying.
 14. In apparatusfor analyzing a waveform, means for simultaneously providing amplitudesamples of said waveform at intervals of ts < or = 1/2W, where ts is theinterval between samples and W is the highest frequency content of saidwaveform, means for simultaneously producing time-varying harmonicallyrelated cosine and sine signals having peak values respectivelyproportional to said samples and having a fundamental frequency which isindependent of the frequency content of said waveform, and means forsimultaneously combining respective ones of said cosine and sine signalsto produce a time-varying spectrum signal having a time variablesimulating the frequency variable of the spectrum of said waveform andhaving a period equal to the fundamental frequency of said time-varyingharmonically related cosine and sine signals.
 15. The invention inaccordance with claim 14, wherein said time-varying harmonically relatedcosine and sine signals are expressable as e1cosw1t, e2cosw2t, . . .encoswnt and e1sinw1t, e2sinw2t . . . encoswnt, where e1, e2, . . . encorrespond to said samples, n is the number of samples, t is the timevariable, and w is frequency in radians per second, and wherein thefrequencies w1 to wn and the fundamental frequency wo thereof satisfythe relatiOnships w1 (b+1)wo, w2 (b+2)wo, . . . wn (b+n)wo, where b isan integer and includes zero.
 16. The invention in accordance with claim14, wherein said spectrum signal corresponds to the amplitude spectrumof said waveform and is expressable as
 17. The invention in accordancewith claim 15, wherein said spectrum signal corresponds to the phasespectrum of said waveform and is expressable as
 18. The invention inaccordance with claim 15, wherein said means for combining said cosineand sine signals produces a time-varying amplitude spectrum signal A''and a time-varying phase spectrum signal phi '' expressable as
 19. Theinvention in accordance with claim 28, wherein said apparatusadditionally includes output means to which said spectrum signal isapplied for providing a manifestation of said waveform.
 20. Theinvention in accordance with claim 19 wherein said output means includescathode ray tube means for producing a display of at least one of theamplitude and phase spectra of said input waveform.
 21. The invention inaccordance with claim 14 wherein said waveform comprises a plurality ofpulses, and wherein said fundamental frequency is chosen to provide ananalysis time less than the time between said pulses.
 22. The inventionin accordance with claim 14, wherein said means for producingtime-varying harmonically related cosine and sine signals includes meansfor producing a first group of time-varying harmonically related cosineand sine signals having peak values respectively corresponding to afirst plurality of said samples, means for producing a second group oftime-varying harmonically related cosine and sine signals of likefrequencies of those of said first group and having amplitudescorresponding to a second plurality of samples, and means for convertingthe frequencies of the second group of cosine and sine signals tofrequencies which continue the harmonic relation of the first group. 23.The invention in accordance with claim 18, wherein said means forcombining includes: means for producing time-varying cosine and sinesignals of like frequency, means for multiplying the signal Ec'' by oneof the cosine and sine signals of like frequency and for multiplying thesignal Es'' by the other, means for summing the thus multiplied signals,and envelope detector means for producing the amplitude spectrum signalA'' from the summed multiplied signals.
 24. The invention in accordancewith claim 19, wherein said means for combining additionally includes:phase detector means to which the summed multiplied signals are appliedalong with one of said cosine and sine signals of like frequency fordetecting the phase difference phi o'' therebetween, and means forsubtracting the term (bts)t from phi o'' for producing the phasespectrum signal phi ''.
 25. In apparatus for analyzing a waveform, abasic unit for providing a plurality of samples of said waveform atintervals satisfying the Nyquist sampling theorem and for producingtime-varying harmonically related cosine and sine signals having peakvalues respectively proportional to said samples, at least one add-onunit for providing an additional plurality of samples of said waveformat Nyquist sampling intervals and for producing time-varyingharmonically related cosine and sine signals having peak valuesrespectively proportional to the additional plurality of samples andwith like frequencies as those produced by said basic unit, said add-onunit also including means for up-converting the frequencies of thecosine and sine signals produced thereby to frequencies such that thecosine and sine signals of the add-on unit will continue the harmonicrelation of the cosine and sine signals of the basic unit, and means forcombining all of said cosine and sine signals to produce a time-varyingspectrum signal of said waveform with the time variable of said spectrumsignal simulating the frequency variable of the spectrum of saidwaveform and said spectrum signal having a period equal to thefundamental frequency of the time-varying harmonically related sine andcosine signals.
 26. The invention in accordance with claim 25, whereinthe cosine and sine signals produced by said basic unit are expressableas e1cosw1t, e2cosw2t . . . encoswnt and e1sinw1t, e2sinw2t . . .ensinwnt, where e1, e2, . . . en are the samples provided by said basicunit, w1 to wn are the harmonically related frequencies of the cosineand sine signals, and n is the number of samples provided by the basicunit, wherein the cosine and sine signals produced by said add-on unitprior to application to said means for up-converting are expressable asen 1cosw1t, en 2cosw2t, . . . en mcoswnt and en 1sinw1t, en 2sinw2t . .. en msinwnt, where en 1, en 2 . . . en m are the samples provided bysaid add-on units, w1 to wn are as defined above, and m is the number ofsamples provided by the add-on unit, wherein the cosine and sine signalsprovided by said add-on unit after application to said means forup-converting are expressable as en 1coswn 1t, en 2coswn 2t . . . enmcoswn mt, and en 1sinwn 1t, en 2sinwn 2t . . . en msinwn mt, andwherein the frequencies w1 to wn and wn 1 to wn m and the fundamentalfrequencies wo thereof satisfy the relationships w1 (b+1)wo, w2 (b+2)wo. . . wn (b+n)wo, wn 1 (b+n+1)wo, wn 2 (b+n+2)wo . . . wn m (b+n+m)wo,where b is an integer including zero.
 27. In a method of analyzing thespectrum of a waveform in essentially real-time, the steps of: amplitudesampling said waveform at intervals of ts < or = 1/2W, where ts is theinterval between samples and W is the highest frequency content of saidwaveform, simultaneously generating time-varying harmonically relatedcosine and sine signals having peak values respectively proportional tosaid samples and expressable as e1cosw1t, e2cosw2t . . . encoswnt ande1sinw1t, e2sinw2t . . . ensinwnt, where e1, e2 . . . en correspond tosaid samples, n is the number of samples, t is the time variable and wis frequency in radians per second, and the frequencies w1 to wn and thefundamental frequency wo thereof satisfy the relationships w1 (b+1)wo,w2 (b+2)wo . . . wn (b+n)wo, where b is an integer and includes zero,and simultaneously combining the generated cosine and sine signals toproduce a time-varying spectrum signal such that the time variable ofsaid spectrum signal simulates the frequency variable of the spectrum ofsaid waveform and said spectrum signal has a period equal to thefundamental frequency wo.
 28. The invention in accordance with claim 27,wherein the step of combining produces an amplitude spectrum signal A''and a phase spectrum signal phi '' expressable as
 29. The invention inaccordance with claim 27, wherein the step of generating is performed bysteps including generating pulse trains having harmonically relatedfrequencies with amplitudes proportional to said samples, and filteringthe resultant signals over narrow frequency bands corresponding to therespective frequencies of said pulse trains.
 30. The invention inaccordance with claim 27, wherein said steps of sampling and generatingare performed by steps including generating a first group oftime-varying harmonically related cosine and sine signals havingamplitudes proportional to a first plurality of said samples, generatinga second group of time-varying harmonically related cosine and sinesignals of like frequencies as those of said first group and havingamplitudes proportional to a second plurality of said samples, andconverting the frequencies of the second group of cosine and sinesignals to frequencies which continue the harmonic relation of the firstgroup.
 31. In a method of deriving an essentially real-time electricalsignal representative of a desired Fourier-type mathematicaltransformation of an input signal, the steps of: sampling said inputsignal to provide fixed value amplitude samples representative thereof,simultaneously generating harmonically related time-varying cosine andsine signals having peak values proportional to respective ones of saidsamples, and simultaneously combining said cosine and sine signals toproduce essentially real-time resultant electrical signalsrepresentative of the desired Fourier-type mathematical transformation.32. The invention in accordance with claim 31, wherein said desiredFourier-type mathematical transformation is the Fourier transform of theinput signal, wherein said sampling is such as to sample said inputwaveform in a manner so that said samples are amplitude samples of saidinput signal taken at intervals satisfying the Nyquist sampling rate,and wherein said resultant electrical signals comprise twotime-simulated electrical signals representative of the real andimaginary components of the Fourier transform of the input signal. 33.The invention in accordance with claim 31, wherein said desiredFourier-type mathematical transformation is the inverse Fouriertransform of the input signal, wherein said input signal is provided inthe forms of time-simulated signals corresponding to its real andimaginary components, and wherein said sampling is such as to separatelysample each of said time-simulated signals in a manner so that saidsamples are amplitude samples thereof taken at sufficiently closelyspaced intervals so that said resultant electrical signals arerepresentative of the inverse Fourier transform.
 34. The invention inaccordance with claim 33, wherein said generating is such that theharmonically related time-varying cosine signals having peak valuesproportional to respective ones of the amplitude samples obtained fromthe time-simulated signal corresponding to the real component of theinput signal and the harmonically related time-varying sine signals havepeak values corresponding to respective ones of the amplitude samplesobtained from the time-simulated signal corresponding to the imaginarycomponent of the input signal.
 35. In a method of performingFourier-type mathematical manipulations on electrical signals, the stepsof: sampling an input signal at least at a Nyquist sampling rate toprovide fixed value amplitude samples representative thereof, generatinga first series of harmonically related time-varying cosine and sinesignals having peak values proportional to respective ones of saidsamples, combining said cosine and sine signaLs to producetime-simulated electrical signals representative of the real andimaginary components of the Fourier transform of said input signal,sampling each of the real and imaginary component signals at least at aNyquist sampling rate to provide fixed value amplitude samples thereof,modifying the fixed value amplitude samples obtained from sampling thereal and imaginary component signals in accordance with desiredcriteria, generating a second series of harmonically relatedtime-varying cosine and sine signals having peak values proportional torespective ones of the modified samples, and respectively summing thesecond series of cosine and sine signals.
 36. The invention inaccordance with claim 35, wherein said method includes the step ofcomparing the fixed value amplitude samples obtained from sampling thereal and imaginary component signals with predetermined referencevalues, and wherein the step of modifying modifies the amplitude samplesobtained from sampling the real and imaginary component signals inresponse to the results obtained during the step of comparing.
 37. Theinvention in accordance with claim 35, wherein said method includes thestep of combining the respectively summed second series of cosine andsine signals to produce a resultant electrical signal representative ofsaid input signal but containing the modifications introduced by thestep of modifying.
 38. In apparatus for deriving an electrical signalrepresentative of a Fourier-type mathematical transformation of an inputsignal, the combination of: sampling means including a plurality ofsample and hold circuits for sampling said input signal and for storingsamples representative thereof in respective ones of saidsample-and-hold circuits, generating means responsive to the samplesstored in said sample-and-hold circuits for generating harmonicallyrelated time-varying cosine and sine signals having peak valuesproportional to respective ones of said samples, said generating meansincluding: means for producing a first group of time-varyingharmonically related cosine and sine signals having peak valuesrespectively corresponding to a first plurality of said samples, meansfor producing a second group of time-varying harmonically related cosineand sine signals of like frequencies of those of said first group andhaving amplitudes corresponding to a second plurality of samples, meansfor converting the frequencies of the second group of cosine and sinesignals to frequencies which continue the harmonic relation of the firstgroup, and means for combining said cosine and sine signals to produceresultant electrical signals representative of the desired Fourier-typemathematical transformation.
 39. In apparatus for deriving an electricalsignal representative of a Fourier transform of an input signal, thecombination of: sampling means including a plurality of sample-and-holdcircuits for sampling said input signal at least at a Nyquist rate andfor storing amplitude samples representative thereof in respective onesof said sample-and-hold circuits, generating means responsive to thesamples stored in said sample-and-hold circuits for generatingharmonically related time-varying cosine and sine signals having peakvalues proportional to respective ones of said samples, saidtime-varying harmonically related cosine and sine signals beingexpressable as e1cosw1t, e2cosw2t, . . . encoswnt and e1sinw1t, e2sinw2t. . . encoswnt, where e1, e2, . . . en correspond to said samples, n isthe number of samples, t is the time variable, and w is frequency inradians per second, and where the frequencies w1 to wn and thefundamental frequency wo thereof satisfy the relationships w1 (b+1)wo,w2 (b+2)wo, . . . wn (b+n)wo, wherein b is an integer and includes zero,and where the fundamental frequency wo of said cosine and sine signalsmay be chosen independently of the frequency content of said inputsignal, and means for combining said cosine and sine signals to produceresultant electrical signals representative of the desired Fourier-typemathematical transformation, said combining means operating torespectively sum said cosine and sine signals so that said resultantsignals comprise two signals corresponding to the real and imaginarycomponents of the Fourier transform of the input signal.
 40. Theinvention in accordance with claim 39, wherein said generating meansincludes means for weighting the peak values of said cosine and sinesignals.
 41. The invention in accordance with claim 39, wherein saidcombining means includes means for combining said resultant signals toprovide a time-varying spectrum signal representative of the amplituedspectrum of the input signal with the time variable simulating thefrequency variable and having a period equal to the fundamentalfrequency wo of said cosine and sine signals.
 42. The invention inaccordance with claim 41, wherein said combining means additionallyincludes means for combining said resultant signals to provide acontinuous time-varying spectrum signal representative of the phasespectrum of the input signal with the time variable simulating thefrequency variable of the phase spectrum and having a period equal tothe fundamental frequency wo of said cosine and sine signals.
 43. Theinvention in accordance with claim 42, wherein said apparatus includesmeans for displaying at least one of the amplitude and phase spectrasignals.
 44. In apparatus for deriving an electrical signalrepresentative of an inverse Fourier transform of an input signalprovided in the form of two time-simulated signals corresponding to itsreal and imaginary components, the combination of: sampling meansincluding a plurality of sample-and-hold circuits for sampling saidtime-simulated signals at least at a Nyquist rate and for storingamplitude samples representative thereof in respective ones of saidsample-and-hold circuits, a separate set of sample-and-hold circuitsbeing provided for the amplitude samples of each of said time-simulatedsignals, generating means responsive to the samples stored in saidsample-and-hold circuits for generating harmonically relatedtime-varying cosine and sine signals having peak values proportional torespective ones of said samples, said time-varying harmonically relatedcosine and sine signals being expressable as e1cosw1t, e2cosw2t, . . .encoswnt and e1sinw1t, e2sinw2t . . . encoswnt, where e1, e2, . . . encorrespond to said samples, n is the number of samples, t is the timevariable, and w is frequency in radians per second, and where thefrequencies w1 to wn and the fundamental frequency wo thereof satisfythe relationships w1 (b+1)wo, w2 (b+2)wo, . . . wn (b+n)wo, where b isan integer and includes zero, and where the fundamental frequency wo ofsaid cosine and sine signals may be chosen independently of thefrequency content of said input signal, and means for combining saidcosine and sine signals to produce resultant electrical signalsrepresentative of the desired Fourier-type mathematical transformation,said combining means operating to respectively sum said cosine and sinesignals so that said resultant signals comprise two signalscorresponding to the real and imaginary components of the inverseFourier transform of the input signal.
 45. The invention in accordancewith claim 44, wherein said generating means includes means forweighting the peak values of said cosine and sine signals.
 46. Theinvention in accordance with claim 45, wherein said combining meansincludes means for combining said resultant signals to provide a singlecontinuous time-varying signal representative of the inverse Fouriertransform of the input signal.